I'm studying Skorokhod space, which consists of cadlag functions, and I encountered the following sentence:
If a metric space $(\mathbb{S}, \mathcal{S}, d)$ is not separable, then functions that map a probability space (Ω,F, P) to $(\mathbb{S}, \mathcal{S}, d)$ often fail to be measurable.
Here, we can substitute $\mathbb{S}$ with $\mathbb{D}$(Skorokhod space) and $d$ with an uniform metric.
However, I don't have any idea how separability affects measurability.
Can anyone give help?